How do I show that $xy$ is continuous?
I know that the product of two continuous functions is continuous but how do I show that $x$ is continuous and $y$ is continuous?
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4Directly from the definition of continuity. When the adversary picks an $\varepsilon$, set $\delta=\varepsilon$, et voilà! – hmakholm left over Monica Feb 22 '15 at 17:04
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^ excellent, stealing – Simon S Feb 22 '15 at 17:12
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2The statement "the product of two continuous function is continuous" is equivalent to "the function $(x,y)\mapsto xy$ is continuous" – Milo Brandt Feb 22 '15 at 17:17
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The function $f(x,y) = x$ is continuous since given $\epsilon > 0$ and $(a,b)\in \Bbb R^2$, setting $\delta = \epsilon$ makes
$$|f(x,y) - f(a,b)| = |x - a| = \sqrt{(x - a)^2} \le \sqrt{(x - a)^2 + (y - b)^2} < \epsilon$$
whenever $\sqrt{(x - a)^2 + (y - b)^2} < \delta$. Similarly, the function $g(x,y) = y$ is continuous.
kobe
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The projection $(x,y)\mapsto x$ is a linear transformation and in finite dimensional space $\Bbb R^2$ it's continuous. The same for the second projection and you know the rest of the story.